Let a + b = 2c, then which of the following expressions is true?

Correct option is C

a + b = 2c

Express a in Terms of b and c

a = 2c – b

Substitute a = 2c - b in the Given Expressions and Verify

Option A: a² + 2ac = b² + 2bc

Substituting a = 2c - b:

(2c - b)² + 2(2c - b)c = b² + 2bc

Expanding:

4c² - 4bc + b² + 4c² - 2bc = b² + 2bc

8c² - 6bc + b² ≠ b² + 2bc

Since the equation is not equal, this is Incorrect.

Option B: a² - 2ac = b² - bc

Substituting a = 2c - b:

(2c - b)² - 2(2c - b)c = b² - bc

Expanding:

4c² - 4bc + b² - 4c² + 2bc = b² - bc

-2bc + b² ≠ b² - bc

Since the equation is not equal, this is Incorrect.

Option C: a² + 2bc = b² + 2ac

Substituting a = 2c - b:

(2c - b)² + 2bc = b² + 2(2c - b)c

Expanding:

4c² - 4bc + b² + 2bc = b² + 4c² - 2bc

4c² - 2bc + b² = b² + 4c² - 2bc

Since both sides are equal, this is Correct.

Option D: a² - ac = b² + 2bc

Substituting a = 2c - b:

(2c - b)² - (2c - b)c = b² + 2bc

Expanding:

4c² - 4bc + b² - 2c² + bc = b² + 2bc

2c² - 3bc + b² ≠ b² + 2bc

Since the equation is not equal, this is Incorrect.

Correct Answer:

C. a² + 2bc = b² + 2ac